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1) If $P\in K[X]$ irreducible, and if $\alpha_1,...,\alpha_n$ are his roots, does his splitting field is $K(\alpha_1,...,\alpha_n)$ ?

2) If $P(X)=P_1(X)\cdot ...\cdot P_n(X)\in K[X]$ where $P_i$ are irreducible and the roots of $P_i(X)$ are $\alpha_1^i,...,\alpha_{m_i}^i$, does $$K(\alpha_1^1,...,\alpha_{m_1}^1,\alpha_1^2,...,\alpha_{m_2}^2,...,\alpha_1^n,...,\alpha_{m_n}^n)$$ is the splitting field of $P(X)$ ?

In both case, I think it's true, but I would like to have a confirmation.

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    $\begingroup$ It's exactly that. $\endgroup$
    – Bernard
    Nov 17, 2015 at 10:09
  • $\begingroup$ You're welcom. Comment aside: avoid these ‘upper indices’, as they may be confused with true exponents. $\endgroup$
    – Bernard
    Nov 17, 2015 at 10:20

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